Playing match-up:

Jon: “the wind cut like a knife up here” Alayne: “the wind was blowing fiercely, as it always did up here”

Jon: “with hanging icicles that looked like long white teeth from a distance” Alayne: “Icicles twenty feet long draped the lip of the precipice”

Jon: “Yet even so, Jon Snow was not sorry he had come. There were wonders here as well.” Alayne: “but the view made her forget all that for half a heartbeat.”

Jon: “the lips of sheer stone cliffs” Alayne: “the lip of the precipice”

Jon: “icy thin waterfalls” Alayne: “the frozen waterfall”

Jon: “sunlight flashing” Alayne: “the sun was hitting the mountains”

Jon: “..and a mountain meadow” Alayne: “..its forests and rivers and fields”

Jon: “piper’s grass in russet and gold.” Alayne: “made them look like solid gold”

Jon: “Eagles nested in the heights.” Alayne: “A falcon soared above”

Jon: “circling effortlessly on great blue-grey wings that seemed almost part of the sky” Alayne: “blue wings spread wide against the morning sky”

Jon: “He wished he could move as sure and silent as that shadowcat, and kill as quickly.” Alayne: “Would that I had wings as well… If only they were truly ants, she thought, we could step on them and crush them”

[Word sequence] Jon: “Snow [...] of sheer stone [...] the sky” Alayne: “The snow [...] of stone [...] the morning sky”

Jon VI : Jon in the mountains, learning how to climb, meeting the wildlings (& Ygritte), getting told the story of Bael taking the Winter Rose. *Note: also followed immediately by a Sansa chapter, and there are several other parallels to her in the passage.

Alayne I : First Alayne chapter, as she is in the Eyrie mountains after she gets taken by Baelish. She meets the lords and is now fully faking her identity in the Vale, as a bastard with the surname Stone.

Jon VI has many allusions and references to Sansa, but particularly Jon's climb here has another mirror parallel to when Sansa was making the climb to flee King's Landing:

“One step and then another, Jon told himself. One step and then another, and I will not fall.” – Jon VI, ASOS

"One more step, she told herself, one more step. She had to keep moving. [...] One more step, and one more step.” "I did it. I did it, I didn’t fall.." – Sansa V, ASOS

Bravery and perseverence on a "climb" is a recurring theme in both of their chapters. In a similar manner to Jon VI, there's also a fun allusion to the castles in another Alayne passage, in the correct order. It's also an affirmation that Alayne Stone, her bastard identity arc, is explicitly tied to the three, and calls back to "bastard-like" bravery (very reminiscent of Jon's):

— Alayne II, AFFC

And Sky, Snow and Stone are also nearby when she remarks on the Vale/Eyrie not being her home, the same chapter that irrevocably marks the North as her one, true home:

— Sansa VII, ASOS

Of course, there's an even more important and infamous thematic tie to stone, snow and castles in Sansa's story - Winterfell. It's remarked several times that Winterfell's strength is its stone, and thanks to it, it stands even after being attacked and broken.

In ASOS, Jon Snow says that he wants to rebuild Winterfell, and in the very next chapter Sansa is echoing this desire by building the castle out of snow:

"Winterfell, he thought. Theon left it burned and broken, but I could restore it." — Jon XII, ASOS

“The snow fell and the castle rose” “It was only a castle, but before very long Sansa knew it was Winterfell.” — Sansa VII, ASOS

Snow and Stone we can easily attribute to the two. But for the last castle,, I believe "Sky" may refer to Bran - in Sansa/Alayne chapters, these three waycastles are tied twice to birdflight and specifically her gaining wings. It's Bran, the Winged Wolf, who wants to teach his family to fly the same way he gained wings.

"I'd tell him I could fly, but he wouldn't believe, so I'd have to show him. I bet that he could learn to fly too, him and Arya and Sansa, even baby Rickon and Jon Snow. We could all be ravens and live in Maester Luwin's rookery." — Bran III, ADWD

It's also Bran, who himself has close ties to the roots of Winterfell, that remarks on the strength of stone keeping the broken castle standing:

"Beyond, the tops of the keeps and towers still stood as they had for hundreds of years, and it was hard to tell that the castle had been sacked and burned at all. The stone is strong, Bran told himself, the roots of the trees go deep, and under the ground the Kings of Winter sit their thrones. So long as those remained, Winterfell remained." — Bran VII, ACOK

Bran was confirmed to be the ruling monarch on the Iron Throne by the end of ASOIAF. Jon and Sansa are very likely, if not virtually guaranteed, to gain leadership positions in their arc as well. They're the three characters with the greatest tie to the restoration of Winterfell, and also the three Stark children with the heaviest "bird" imagery, which they're grouped by in the text: